a genuinely high order total variation diminishing …zhan1966/research/paper/tvd.pdftvd or tvb...
TRANSCRIPT
A GENUINELY HIGH ORDER TOTAL VARIATION DIMINISHING
SCHEME FOR ONE-DIMENSIONAL SCALAR CONSERVATION
LAWS∗
XIANGXIONG ZHANG† AND CHI-WANG SHU‡
Abstract. It is well known that finite difference or finite volume total variation diminishing(TVD) schemes solving one-dimensional scalar conservation laws degenerate to first order accuracy atsmooth extrema [8], thus TVD schemes are at most second order accurate in the L1 norm for generalsmooth and non-monotone solutions. However, Sanders [12] introduced a third order accurate finitevolume scheme which is TVD, where the total variation is defined by measuring the variation ofthe reconstructed polynomials rather than the traditional way of measuring the variation of the gridvalues. By adopting the definition of the total variation for the numerical solutions as in [12], it ispossible to design genuinely high order accurate TVD schemes. In this paper, we construct a finitevolume scheme which is TVD in this sense with high order accuracy (up to sixth order) in the L1
norm. Numerical tests for a fifth order accurate TVD scheme will be reported, which include testcases from traffic flow models.
Key words. hyperbolic conservation laws; finite volume scheme; total variation diminishing;total variation bounded; high order accuracy; conservative form
AMS subject classifications. 65M06
1. Introduction. We consider numerical solutions of one-dimensional hyper-bolic scalar conservation law
(1.1) ut + f(u)x = 0, u(x, 0) = u0(x),
where u0(x) is assumed to be a bounded variation function. The main difficultyin solving (1.1) is that the solution may contain discontinuities even if the initialcondition is smooth.
Successful numerical schemes for solving (1.1) are usually total variation stable,for example the total variation diminishing (TVD) schemes [1] or the total variationbounded (TVB) schemes [14], or essentially non-oscillatory, for example the essentiallynon-oscillatory (ENO) schemes [2, 15] or the weighted ENO (WENO) schemes [6,4]. ENO and WENO schemes, although uniformly high order accurate and stablein applications, do not have mathematically provable TVB properties for generalsolutions and do not satisfy a maximum principle. It is certainly desirable to have aTVD or TVB scheme, which shares the TVD property of the exact entropy solutionof (1.1), satisfies a maximum principle, and has at least a convergent subsequence toa weak solution of (1.1) due to its compactness.
Typically, for a finite difference scheme with the numerical solution given by thegrid values uj , or a finite volume scheme with the numerical solution given by the cellaverages uj , the total variation of the numerical solution is measured by
(1.2) TV (u) =∑
j
|uj+1 − uj |
∗Research supported by NSF grant DMS-0809086 and AFOSR grant FA9550-09-1-0126.†Department of Mathematics, Brown University, Providence, RI 02912. E-mail:
[email protected]‡Division of Applied Mathematics, Brown University, Providence, RI 02912. E-mail:
1
which is the standard bounded variation semi-norm when the numerical solution isconsidered to be a piecewise constant function with the data uj . A scheme is TVD ifthe numerical solution satisfies TV (un+1) ≤ TV (un) where un refers to the numericalsolution at the time level tn. A TVB scheme is one which satisfies TV (un) ≤ M forall n such that tn ≤ T , where the constant M does not depend on the mesh sizes butmay depend on T . A sufficient condition for a scheme to be TVB is
TV (un+1) ≤ TV (un) + M∆t, or TV (un+1) ≤ (1 + M∆t)TV (un),
where M is a constant and ∆t is the time step.
It is well known that finite difference or finite volume TVD schemes solving (1.1),where the total variation is measured by (1.2), necessarily degenerate to first orderaccuracy at smooth extrema [8], thus TVD schemes are at most second order accuratein the L1 norm for general smooth and non-monotone solutions. While the TVBschemes in [14] can overcome this accuracy degeneracy difficulty, the schemes are nolonger scale-invariant (scale-invariance refers to the fact that the scheme does notchange when x and t are scaled by the same factor) and involve a TVB parameter Mwhich must be estimated and adjusted for individual problems.
In [12], Sanders introduced a third order accurate finite volume scheme which isTVD. The main idea in [12] is to define the total variation by measuring the variationof the reconstructed polynomials, rather than the traditional measurement as in (1.2).The scheme of Sanders in [12] can be summarized in the following steps.
• Start from the cell averages u0j and the cell boundary values u0
j+ 12
for all j
from the initial condition u0(x).• For n = 0, 1, · · · , perform the following
1. Reconstruct a piecewise quadratic polynomial solution un(x), based onthe information un
j and unj+ 1
2
for all j, such that un(x) is third order ac-
curate (degenerates to second order at isolated critical points, thereforestill third order in the L1 norm), and TVD
(1.3) TV (un(x)) ≤ TV (un−1(x))
where the total variation is measured by the standard bounded variationsemi-norm of the piecewise quadratic polynomial solution un(x). Forn = 0, TV (un−1(x)) is taken as the bounded variation semi-norm of theinitial condition u0(x).
2. Evolve the PDE (1.1) exactly for one time step ∆t from the “initialcondition” un(x) at the time level tn, and take the cell averages un+1
j
and the cell boundary values un+1j+ 1
2
for all j from this exactly evolved
solution. Then return to Step 1 above.
The crucial step in Sanders’ scheme is the reconstruction, which should be highorder accurate and TVD in the sense of (1.3). The second step above, namely theexact time evolution and cell averaging, does not increase the total variation. Theresulting scheme is thus TVD as long as the reconstruction is TVD. The purposeof this paper is to generalize the scheme of Sanders, mainly the step of the highorder TVD reconstruction, to higher order accuracy. In order to measure the totalvariation of a polynomial p(x) of degree k over the interval [a, b], we would need toobtain the zeros of its derivative p′(x) in this interval, denoted by a1, a2, · · · , ak−1. Ifwe also denote a0 = a and ak = b, then the variation of p(x) over the interval [a, b] is
2
∫ b
a |p′(x)|dx =k∑
j=1
|p(aj)− p(aj−1)|. If we insist on working with explicit formulas for
the zeros of polynomials to save cost, then we can only have the polynomial degree kof p(x) up to five, hence our approach can generate schemes up to sixth order in theL1 norm.
A major difficulty in generalizing Sanders’ TVD reconstruction to higher order isthe design of the nonlinear limiter, which should maintain accuracy in smooth regionswhile enforcing the TVD property (1.3). The original limiter of Sanders in [12] workswell for the third order scheme (k = 2), but it seems difficult to generalize it directlyto higher order. We will develop a different limiter to achieve this purpose. The finitevolume schemes we develop in this paper are (k+1)-th order accurate in the L1 norm,conservative, and TVD in the sense of (1.3). Numerical tests for a fifth order accurateTVD scheme will be reported, which include test cases from traffic flow models.
This paper is divided into six sections. In Section 2 we develop a Hermite typereconstruction procedure using piecewise polynomials of degree k for functions ofbounded variation. We use a quartic polynomial (k = 4) as an example to illustratethe procedure. The reconstruction does not increase the variation of the functionbeing approximated, and is (k + 1)-order accurate in regions where the approximatedfunction is smooth. In Section 3 we show how to evolve the approximation in timein essentially the same way as in [12]. In Section 4, we derive the conservative formof the scheme and show that the local truncation error is formally (k + 1)-th orderaccurate. In Section 5 we present some numerical examples of the fifth order schemeusing quartic reconstruction polynomials. Numerical examples include those fromtraffic flow models. Finally, in Section 6, we give concluding remarks and remarks forfuture work.
2. A fifth order TVD reconstruction. For a smooth function u(x) withbounded variation over an interval J ⊂ R, we would like to find a piecewise polyno-mial r(x) approximating u(x) with the property that the variation of r(x) does notexceed that of u(x). We will use a Hermite type quartic reconstruction polynomialas an example to show how to find such an approximation. However we remark thatthis approximation procedure can also be applied to any reconstruction polynomialsof degree up to five, including those obtained with the ENO or WENO procedure.
We will use the following notation for our mesh Ij = [xj− 12, xj+ 1
2], xj =
12
(xj− 1
2+ xj+ 1
2
), ∆x = xj+ 1
2−xj− 1
2, which is assumed to be uniform for simplicity,
and for the discretization of u over the mesh, uj = 1∆x
∫Ij
u(x) dx, uj− 12
= u(xj− 12).
Given the cell averages and cell boundary point values of u(x), there are many waysto obtain a reconstruction polynomial pj(x) over the cell Ij . We choose to use theHermite type reconstruction of degree four; i.e., the polynomial pj(x) should satisfy
1
∆x
∫
Ii
pj(x) dx = ui, i = j − 1, j, j + 1; and pj(xj± 12) = u(xj± 1
2).
If the reconstruction is written as pj(x) = a4(x − xj)4 + a3(x − xj)
3 + a2(x − xj)2 +
a1(x − xj) + a0, then the coefficients can be given explicitly as
a0 =uj−1+298uj+uj+1−54(u
j− 12
+uj+ 1
2
)
192 , a1 =uj−1−uj+1−10(u
j− 12
−uj+ 1
2
)
8∆x ,
a2 =−(uj−1+58uj+uj+1)+30(u
j− 12
+uj+ 1
2
)
8∆x2 , a3 =uj+1−uj−1+2(u
j− 12
−uj+ 1
2
)
∆x3 ,
a4 =5uj−1+50uj+5uj+1−30(u
j− 12
+uj+ 1
2
)
12∆x4 .
3
It is clear that the following properties hold for pj(x):I Accuracy (recall that u(x) is a smooth function)
(2.1) pj(x) = u(x) + O(∆x5), ∀x ∈ Ij
II Agreement of the cell averages
(2.2)1
∆x
∫
Ij
pj(x)dx = uj , ∀j
Define the piecewise polynomial p(x) =∑
j pj(x)χj(x), where χj is the charac-teristic function on Ij . Notice that for the specific Hermite polynomial pj(x) definedabove, we have p(xj− 1
2) = pj(xj− 1
2) = pj−1(xj− 1
2) = u(xj− 1
2), that is, p(x) is a
continuous function. For any piecewise polynomial function r(x) =∑
j rj(x)χj(x),where rj(x) is a polynomial which may not satisfy rj(xj− 1
2) = u(xj− 1
2) or rj(xj+ 1
2) =
u(xj+ 12), we define the variation of r on each cell by
V ar(rj) =
∫ xj+ 1
2
xj− 1
2
|r′j(x)| dx + |rj(xj− 12) − uj− 1
2| + |rj(xj+ 1
2) − uj+ 1
2|,
and define the variation of r on the whole domain by V ar(r) =∑
j V ar(rj). Thestandard total variation seminorm of r(x) is
TV (r) =∑
j
∫ xj+ 1
2
xj− 1
2
|r′j(x)| dx + |rj(xj− 12) − rj−1(xj− 1
2)|
≤
∑
j
V ar(rj).
The approximation r(x) is TVD if it satisfies TV (r) =∫
J |r′(x)| dx ≤∫
J |u′(x)| dx =TV (u), where the integral is in the generalized sense since r′(x) may contain δ-functions, because r(x) is a piecewise polynomial function which may not be continu-ous at cell interfaces. Obviously, r is a TVD approximation if V ar(rj ) ≤
∫Ij|u′(x)| dx,
∀j, where we assume the cell boundary values u(xj− 12) are well defined and are shared
by both cells Ij−1 and Ij . For convenience, we denote the total variation of the func-tion u over the interval Ij by TV (u)j =
∫Ij|u′(x)| dx.
Now the question is, given a piecewise approximation polynomial p(x) and certaininformation of u(x) including its extrema, its cell boundary values uj− 1
2, its cell
averages uj and its cell total variation TV (u)j for all j, whether we can find a TVDapproximation r(x) by limiting p(x) such that the accuracy condition (2.1) and theconservation condition (2.2) still hold.
To obtain the TVD property, it suffices to enforce
(2.3) V ar(rj) ≤ TV (u)j , ∀j.
We now discuss the limiting process case by case:Case I. pj(x) is monotone in the cell Ij .For this case, the TVD requirement (2.3) is already satisfied since pj(x) is a
Hermite type polynomial which interpolates u at the cell boundaries.Case II. pj(x) is not monotone in the cell Ij , but u(x) is monotone in this cell.In this case, the TVD property (2.3) does not hold for pj(x) because
V ar(pj) > |uj+ 12− uj− 1
2| = TV (u)j .
4
Without loss of generality, we only discuss the situation that u(x) is monotonicallyincreasing in Ij , i.e., uj− 1
2≤ uj+ 1
2and u′(x) ≥ 0, ∀x ∈ Ij . Since pj(x) is not
monotone, we must have p′j(x) < 0 for some x ∈ Ij . Consider pj(x) = pj(x)−αj(x−xj), αj = minx∈Ij
p′j(x). We then have the following lemma.Lemma 2.1. The accuracy condition (2.1) and the cell average agreement condi-
tion (2.2) still hold for pj(x).Proof. The agreement of the cell average is obvious. To prove accuracy, it suffices
to show αj = O(∆x4).We have p′j(x) − u′(x) = O(∆x4) since pj(x) is a fourth degree Hermite approx-
imation of u(x). If p′j(x) attains its minimum at the point xmin ∈ Ij , then we have
u′(xmin) ≥ 0, αj = p′j(xmin) < 0, therefore |αj | = −αj ≤ u′(xmin) − αj = O(∆x4).
We now enforce the TVD requirement (2.3). For this purpose, we apply thefollowing scaling to pj(x) around its cell average on the cell Ij :
rθj
j (x) = θj(pj(x) − uj) + uj ,
where θj ∈ [0, 1] is a parameter to be determined by the TVD requirement (2.3).Lemma 2.2. If V ar(pj) > TV (u)j , then there exists θj ∈ [0, 1] such that
V ar(rθj
j ) = TV (u)j .Proof. Take
θj = min
{∣∣∣∣∣uj− 1
2− uj
pj(xj− 12) − uj
∣∣∣∣∣ ,
∣∣∣∣∣uj+ 1
2− uj
pj(xj+ 12) − uj
∣∣∣∣∣
}.
First, we need to show θj ∈ [0, 1]. If θj > 1, then
(2.4) |uj− 12− uj | > |pj(xj− 1
2) − uj |, |uj+ 1
2− uj | > |pj(xj+ 1
2) − uj |.
Since u(x) and pj(x) are both monotonically increasing in Ij , we have
(2.5) uj− 12≤ uj ≤ uj+ 1
2
and pj(xj− 12) ≤ uj ≤ pj(xj+ 1
2). Therefore, (2.4) implies uj− 1
2≤ pj(xj− 1
2) ≤ uj ≤
pj(xj+ 12) ≤ uj+ 1
2. We then have
V ar(pj) =
∫ xj+ 1
2
xj− 1
2
|p′j(x)| dx + |pj(xj− 12) − uj− 1
2| + |pj(xj+ 1
2) − uj+ 1
2|
= (pj(xj+ 12) − pj(xj− 1
2)) + (pj(xj− 1
2) − uj− 1
2) + (uj+ 1
2− pj(xj+ 1
2))
= uj+ 12− uj− 1
2= TV (u)j .
This contradicts V ar(pj) > TV (u)j , therefore we have θj ∈ [0, 1].Second, we should verify the TVD property (2.3). For convenience, we denote
rθj
j (x) by rj(x) here. By the definition of the θj ,
|rj(xj− 12) − uj | = θj |pj(xj− 1
2) − uj | ≤ |uj− 1
2− uj |,
|rj(xj+ 12) − uj | = θj |pj(xj+ 1
2) − uj | ≤ |uj+ 1
2− uj |.(2.6)
Since u(x) and rj(x) are both monotonically increasing in Ij , we have (2.5)and rj(xj− 1
2) ≤ uj ≤ rj(xj+ 1
2), therefore (2.6) implies uj− 1
2≤ rj(xj− 1
2) ≤ uj ≤
5
rj(xj+ 12) ≤ uj+ 1
2. We thus have
V ar(rj) =
∫ xj+ 1
2
xj− 1
2
|r′j(x)| dx + |rj(xj− 12) − uj− 1
2| + |rj(xj+ 1
2) − uj+ 1
2|
= (rj(xj+ 12) − rj(xj− 1
2)) + (rj(xj− 1
2) − uj− 1
2) + (uj+ 1
2− rj(xj+ 1
2))
= uj+ 12− uj− 1
2= TV (u)j .
Notice that this scaling does not change the cell average. Of course, we wouldneed to show that, with the θj taken in Lemma 2.2, this scaling does not destroyaccuracy. We would need the following result for polynomials:
Lemma 2.3. If a polynomial p(x) of degree k (k ≤ 4) is monotone over an intervalI = [a, b], and its cell average over I is p, then
(2.7) maxx∈I
∣∣∣∣p(x) − p
p(b) − p
∣∣∣∣ ≤ C, maxx∈I
∣∣∣∣p(x) − p
p(a) − p
∣∣∣∣ ≤ C,
where C is a constant depending only on the degree of the polynomial k. In particular,C(2) = 2, C(3) = 4 and C(4) = 7.
The proof is elementary but lengthy, and is therefore deferred to the Appendix.A similar (and more general) result for quadratic polynomials can be found in theappendix of [5].
Using Lemma 2.3 we can prove the accuracy property:
Lemma 2.4. For the θj chosen in Lemma 2.2, the accuracy property (2.1) holds
for rθj
j (x).
Proof. It suffices to show that rθj
j (x) − pj(x) = O(∆x5). Without loss of gen-
erality, we assume θj =
∣∣∣∣u
j− 12
−uj
epj (xj− 1
2
)−uj
∣∣∣∣ . In this case, since both u(x) and pj(x) are
monotonically increasing, we actually have θj =u
j− 12
−uj
epj(xj− 12
)−uj. Then,
rθj
j (x) − pj(x) = θj(pj(x) − uj) + uj − pj(x) = (θj − 1)(pj(x) − uj)
=
(u
j− 12
−uj
epj(xj− 12
)−uj− 1
)(pj(x) − uj) =
uj− 1
2
−epj(xj− 12
)
epj(xj− 12
)−uj(pj(x) − uj)
=epj(x)−uj
epj(xj− 12
)−uj(uj− 1
2− pj(xj− 1
2)) = O(∆x5)
where in the last equality we have used Lemma 2.3.
Case III. Neither pj(x) nor u(x) is monotone in Ij .
First, consider the situation that u(x) has only one nontrivial extremum in Ij .Without loss of generality, we assume u(x) attains its maximum at xmax ∈ Ij , and wedenote umax = u(xmax).
We propose to break the polynomial pj(x) into two parts, plj and pr
j . Let ulj and
urj be the cell averages of u on I l
j = [xj− 12, xmax] and Ir
j = [xmax, xj+ 12] respectively,
similarly let plj and pr
j be the cell averages of pj(x) on I lj and Ir
j respectively. Define
plj(x) = pj(x), ∀x ∈ I l
j , prj(x) = pj(x), ∀x ∈ Ir
j ,
ul(x) = u(x), ∀x ∈ I lj , ur(x) = u(x), ∀x ∈ Ir
j .
6
After an adjustment of the cell averages over the two sub-cells, in each of thetwo intervals I l
j and Irj , the situation reduces to either Case I or Case II. Here, we
will only briefly discuss plj , since the procedure for pr
j is the same. Since umax is the
maximum, ul(x) is monotonically increasing in I lj .
1. If plj is monotone on I l
j (Case I), then define
(2.8) plj(x) = pl
j(x) − plj + ul
j , (rlj)
θj (x) = θj(plj(x) − ul
j) + ulj ,
where
(2.9) θj = min
{∣∣∣∣∣uj− 1
2− ul
j
pj(xj− 12) − ul
j
∣∣∣∣∣ ,∣∣∣∣∣
umax − ulj
pj(xmax) − ulj
∣∣∣∣∣ , 1}
.
2. If plj is not monotone on I l
j (Case II), let α be the minimum of the derivative
of plj over I l
j , then α < 0. Define(2.10)
plj(x) = pl
j(x)−plj +ul
j −α
(x −
xj− 1
2
+xmax
2
), (rl
j)θj (x) = θj(p
lj(x)−ul
j)+ulj ,
where
(2.11) θj = min
{∣∣∣∣∣uj− 1
2− ul
j
pj(xj− 12) − ul
j
∣∣∣∣∣ ,∣∣∣∣∣
umax − ulj
pj(xmax) − ulj
∣∣∣∣∣ , 1}
.
We define rrj (x) on Ir
j in a similarly way, and use rj(x) = rlj(x)χ(I l
j )+ rrj (x)χ(Ir
j )as our TVD approximation of u(x) on Ij . By the same arguments as in Cases I andII, we can show the TVD property and agreement of cell averages for rj(x). And wecan also show the accuracy for rj(x), following the same arguments as in Cases I andII.
If there are multiple nontrivial extrema of u(x) inside the interval Ij , for theimplementation of the limiter, we will choose arbitrarily one of the extrema of u(x)(e.g. the one which gives the maximum or the minimum of u(x) inside the intervalIj). Without loss of generality, we assume u(x) attains its maximum at xmax ∈ Ij ,and umax = u(xmax) is chosen as the extremum which will be used in the limiter. Westill break the polynomial pj(x) into two parts, pl
j(x) on I lj = [xj− 1
2, xmax] and pr
j (x)
on Irj = [xmax, xj+ 1
2]. Then we perform the following modification for pl
j(x):
1. If(uj− 1
2− ul
j
)(ul
j − umax)≥ 0 and pl
j(x) is monotonically nondecreasing on
I lj , then do exactly the same limiting as (2.8) and (2.9).
2. If(uj− 1
2− ul
j
)(ul
j − umax)≥ 0 and pl
j(x) is not monotone on I lj , do exactly
the same limiting as (2.10) and (2.11).3. Otherwise, set pl
j(x) = ulj .
The limiting for prj(x) is similar. By the same arguments as above, we can show
the TVD property and the agreement of cell averages for such modified two-piecepolynomials.
Remark 2.5. One apparent gap in the proof above is that accuracy for CaseIII can only be shown to hold for the case of u(x) having at most one extremum inthe cell Ij . We would like to justify this restriction for sufficiently small ∆x. Forthis purpose, we assume that the initial condition u(x, 0) has only finitely many strict
7
smooth extrema. A pair of adjacent extrema can only consist of one maximum Mj
and one minimum mj , if the function u(x, 0) is smooth and non-constant in between,with mj < Mj . Denoting C = minj(Mj − mj) > 0. If we agree to consider accuracyonly in those cells in which | ∂
∂xu(x, t)| ≤ M for a pre-determined constant M , we
will take the mesh size ∆x < CM . If at a later time t, u(x, t) has two extrema in
the same cell, then they must correspond to one such pair in the initial condition withvalues mj and Mj (following characteristics, along which the solution stays constant).Clearly, by the mean value theorem, there is then a point ξ in this cell such that| ∂∂xu(ξ, t)| ≥
Mj−mj
∆x ≥ C∆x > M , hence we do not need to consider accuracy of the
numerical approximation in this cell. In summary, the reconstruction polynomialsafter the modifications introduced in this section always satisfy the TVD and cellaverage agreement properties, however accuracy can be shown only for sufficientlysmall ∆x.
3. A TVD finite volume scheme. Combining the high order accurate TVDreconstruction described in the previous section with the method of characteristics,we obtain a high order accurate TVD Godunov-type finite volume scheme solving theone dimensional scalar conservation law (1.1).
3.1. Time evolution. To implement the approximation of the previous sec-tion, several pieces of information must be available for each cell. Specifically, thecell averages and the left and right cell boundary point values of the function beingapproximated must be known. To obtain this information, we follow Sanders [12] anduse a staggered spatial mesh with the method of characteristics. We only need todiscuss the evolution procedure for one time step.
Let T (u0)(x, t), t ≥ 0, denote the solution to the scalar conservation law (1.1) andR(u0)(x) denote the piecewise polynomial TVD reconstruction to u0(x) in the previ-ous section. As in the previous section, we partition the real line into nonoverlappingintervals Ij = [xj− 1
2, xj+ 1
2], and approximate u0 ∈ BV by the piecewise polynomial
TVD reconstruction
u0(x) = R(u0)(x) =∑
j
rj(x)χj(x),
where rj(x) is either a polynomial or a (possibly discontinuous) two-piece poly-nomials rj(x) = rl
j(x)χ(I lj ) + rr
j (x)χ(Irj ). Consider a staggered partition Ij− 1
2=
[xj−1, xj ]. The objective of the evolution is to determine the necessary informationof T (u0)(x, ∆t
2 ) such that a TVD piecewise polynomial reconstruction at time t = ∆t2
can be obtained.We assume that u0 is given by u0(x) = R(u0)(x), where R denotes a “precon-
ditioned” version of reconstruction R. By this we mean specifically that R(u0) ismodified (in a way we will discuss later) so that for all j,
(3.1) maxIj
∣∣∣∣d
dxu0(x)
∣∣∣∣∣∣f ′′(u0(x))
∣∣ ∆t < 2.
Essentially, to enforce (3.1) is to push extremely large gradients of rj(x) out of theinterval Ij and into the jump discontinuities at cell interfaces, see [12]. An additionalcondition that we assume throughout is the Courant condition; that is, for all u inthe range of u0 we assume the ratio λ = ∆t/∆x is taken so that
(3.2) |f ′(u)|λ < 1.
8
x
t
�
�
��
xj− 1
2
yj xj+ 1
2
xj
tn
tn+1
2
1
(a) The backward characteristic line from
(xj , tn+ 1
2 ).
s
H(s)
� �� � �
xj− 1
2
xj+ 1
2x0j
s1 s2
(b) A sketch of H(s) in Case 3.
Fig. 3.1.
If we can find the backward characteristic line from each point xj at time t = ∆t/2back to the time level t = 0 with its foot located at yj ∈ Ij (see Figure 3.1(a)) suchthat the exact entropy solution is constant uj = u0(yj) along the line, then we havealready obtained the cell endpoint values of T (u0)(x, ∆t
2 ) on the staggered cell Ij− 12,
and we can apply the divergence theorem to (1.1) over the trapezoid defined by thepoints (xj−1, ∆t/2), (xj , ∆t/2), (yj−1, 0) and (yj , 0) to obtain the cell average ofT (u0)(x, ∆t
2 ) on Ij− 12
as
(3.3)
u12
j− 12
= 1∆x
[∫ yj
yj−1u0(x)dx − ∆t
2 f(uj) + (xj − yj)uj + ∆t2 f(uj−1) − (xj−1 − yj−1)uj−1
].
Therefore, we may now focus on how to obtain the backward characteristic line.With (3.1) and (3.2) we have:Lemma 3.1. If the approximation function u0 is continuous on Ij , that is, rj(x)
is a polynomial, then the backward characteristic equation
(3.4) f ′(u0(x)) =xj − x
∆t/2, x ∈ Ij
has a unique solution yj ∈ Ij .Proof. Consider finding the root of the function
(3.5) H(s) = f ′(u0(s))∆t + 2(s − xj).
According to (3.2), we have that
(3.6) H(x+j− 1
2
) = f ′(u0(x+j− 1
2
))∆t − ∆x < 0,
(3.7) H(x−
j+ 12
) = f ′(u0(x−
j+ 12
))∆t + ∆x > 0.
Moreover, (3.1) implies that for every s ∈ Ij ,
(3.8)d
dsH(s) = f ′′(u0(s))∆t
d
dsrj(s) + 2 > 0.
9
Therefore, H(s) has a unique root yj ∈ Ij = (xj− 12, xj+ 1
2).
Lemma 3.2. If rj(x) = rlj(x)χ(I l
j) + rrj (x)χ(Ir
j ) where rlj(x) and rr
j (x) are two
polynomials defined on I lj = [xj− 1
2, x0
j ] and Irj = [x0
j , xj+ 12] respectively, we can get
either one or three possible candidates for the backward characteristic line from (xj , t12 )
by solving the characteristic equation (3.4).Proof. Again, consider finding the root of the function (3.5). We still have
(3.6) and (3.7). Moreover, (3.8) holds for each piece of s ∈ (xj− 12, x0
j ) and s ∈
(x0j , xj+ 1
2), respectively. Therefore, H(s) is continuous on Ij except at x0
j and H(s)
is monotonically increasing both on the left side and on the right side of x0j . Thus,
there are three cases depending on the signs of H((x0j )
−) and H((x0j )
+).
1. If H((x0j )
−) H((x0j )
+) > 0, there is exactly one root of H(s).
2. If H((x0j )
−) ≤ 0, H((x0j )
+) ≥ 0, the backward characteristic line is from
(x0j , 0) to (xj , ∆t/2) and the speed of this characteristic is 2(xj − x0
j )/∆t.
Precisely speaking there is no root of H(s) if H((x0j )
−) < 0 and H((x0j )
+) > 0,
but this means a rarefaction wave emanating from (x0j , 0).
3. If H((x0j )
−) ≥ 0, H((x0j )
+) ≤ 0, then there are two roots of H(s), one is
in I lj and the other one is in Ir
j . Moreover, the line segment from (x0j , 0) to
(xj , ∆t/2) is also a possible characteristic. See Figure 3.1(b).From the previous lemma, we know that there is a unique backward characteristic
line unless Case 3 happens. For Case 3, we can use the Lax formula to choose thecorrect characteristic line among the candidates, if the flux f(u) is convex, see, e.g. theprocedure used in [10]. However, if f(u) is nonconvex, it is very difficult to single outthe correct characteristic line among these candidates. We will show in next sectionthat any choice among these candidates will maintain the desired accuracy, thereforeour main concern is the TVD property of the scheme. We will choose the candidateso that the cell average obtained from (3.3) satisfies the maximum principle, i.e.
(3.9) min[yj−1,yj ]
u0(x) ≤ u12
j− 12
≤ max[yj−1,yj ]
u0(x).
Once this maximum principle is satisfied, the conditions for the analysis in Section 2will be fulfilled and we may have a TVD reconstruction.
Notice that a choice of the correct characteristics for both cell end points yj−1
and yj would produce u12
j− 12
from (3.3) as the exact cell average of the entropy solu-
tion T (u0)(x, ∆t/2) on Ij− 12, therefore it would automatically satisfy the maximum
principle (3.9). Thus, in Case 3 when there are more than one candidate for the char-
acteristic line, there is at least one choice which would return a u12
j− 12
satisfying (3.9).
If more than one candidate satisfies this criterion, we will simply choose one of themarbitrarily. In our implementation, if there are N intervals in which Case 3 happens,then we have 3N candidates of possible combination. We check all the 3N candidates
sequentially until we find one choice such that each u12
j− 12
from (3.3) satisfies (3.9),
then we will stop the search and use this choice.After finding the backward characteristics, we can obtain the cell averages and
cell boundary point values of T (u0)(x, ∆t/2). With this information, we can apply theprocedure in the previous section to obtain a piecewise Hermite type reconstructionpolynomial p(x) =
∑j pj− 1
2(x)χj− 1
2(x) on the staggered mesh Ij− 1
2= [xj−1, xj ].
To find a TVD approximation to T (u0)(x, ∆t/2), we still need to know the ex-
10
trema of T (u0)(x, ∆t/2). In actual implementation, we use the maximum/minimumvalue of u0(x) on the interval [yj−1, yj ] to replace the maximum/minimum value ofT (u0)(x, ∆t/2), and the position of the extremum of T (u0)(x, ∆t/2) is determinedby forward characteristic lines starting from the maximum/minimum point of u0(x).For example, if xmax
j is the point where the maximum value of u0(x) in [yj−1, yj ] is
achieved, and the maximum value is u0max, then the forward characteristic is the
line through (xmaxj , 0) and (xmax
j + f ′(u0max)∆t/2, ∆t/2), i.e., we trace the maxi-
mum/minimum values by the characteristics. The two cell averages of the left andright parts can be computed in the same way as in (3.3).
Remark 3.3. Since the entropy solution to (1.1) satisfies the maximum principle,the maximum/minimum value of T (u0)(x, ∆t/2) on the interval Ij− 1
2is less/greater
than or equal to the maximum/minimum value of u0(x) on the interval [yj−1, yj ],which could be easily computed because u0 is a piecewise polynomial of degree k ≤ 5and there are algebraic analytical formulas for the extrema.
3.2. Precondition. We propose the following procedure as the preconditioningprocess to ensure (3.1). Clearly, this preconditioning does not affect accuracy insmooth cells.
1. rj(x) is a polynomial. If rj(x) fails to satisfy (3.1), i.e.,
maxx∈Ij
r′j(x)f ′′(rj(x)) > 2/∆t, or minx∈Ij
r′j(x)f ′′(rj(x)) < −2/∆t,
then we use rj(x) = µ(rj(x) − uj) + uj to replace rj(x), where uj is the cellaverage and
µ = min
{∣∣∣∣∣2
∆t maxx∈Ij
f ′′(rj(x)) maxx∈Ij
r′
j(x)
∣∣∣∣∣ ,∣∣∣∣∣
2∆t min
x∈Ij
f ′′(rj(x)) minx∈Ij
r′
j(x)
∣∣∣∣∣
}.
2. rj(x) = rlj(x)χ(I l
j) + rrj (x)χ(Ir
j ). We only discuss rlj(x), the process for rr
j (x)
is similar. If rlj(x) fails to satisfy (3.1), i.e.,
maxx∈Il
j
(rlj
)′(x)f ′′(rj(x)) > 2/∆t, or min
x∈Ilj
(rlj
)′(x)f ′′(rj(x)) < −2/∆t,
then we use rlj(x) = µ(rl
j(x) − ulj) + ul
j to replace rlj(x), where ul
j is the cell
average on I lj and
µ = min
∣∣∣∣∣∣2
∆t maxx∈Ij
f ′′(rlj(x)) max
x∈Ilj
(rlj)
′(x)
∣∣∣∣∣∣,
∣∣∣∣∣∣2
∆t minx∈Il
j
f ′′(rlj(x)) min
x∈Ij(rl
j)′(x)
∣∣∣∣∣∣
.
3.3. Algorithm flowchart. We can now formulate the algorithm flowchart forthe TVD finite volume scheme. From now on we restrict ourself to the fifth order casefor easy presentation, although the procedure and results are valid for all orders up tosix. Given the cell averages and cell boundary values of the initial data u0(x), we usethe Hermite reconstruction procedure in Section 2 to obtain a piecewise polynomialof degree four u0(x) as the numerical initial condition. We apply the limiting processin Section 2 to the numerical initial condition, still denoted by u0(x), which is then apiecewise polynomial of degree four satisfying TV (u0) ≤ TV (u0).
11
1. Start with piecewise polynomial of degree four un(x) =∑
j rj(x)χj(x) attime level n, where rj(x) is either a polynomial or two-piece polynomials onthe interval Ij . Apply the preconditioning process detailed in Section 3.2 toeach rj(x), still denoted as rj(x).
2. For each j, by Lemmas 3.1 and 3.2, there is either one or three solutions inIj for the characteristic equation f ′(rj(x)) =
xj−x∆t/2 , x ∈ Ij . Use either explicit
formulas (if f ′(rj(x)) is a lower degree polynomial) or an iterative methodto find the solution or solutions in Ij . If there is only one solution, set it asyj . Next, if there are N intervals in which there are three candidates for yj ,then check all the 3N candidates of combination sequentially until we find
one which returns a maximum-principle-satisfying un+ 1
2
j− 12
for all the intervals
in the formula
un+ 1
2
j− 12
= 1∆x
[∫ yj
yj−1un(x)dx − ∆t
2 f(un(yj)) + (xj − yj)un(yj)
+∆t2 f(un(yj−1)) − (xj−1 − yj−1)u
n(yj−1)].(3.10)
3. Evaluate T (un)(xj ,∆t2 ) = un(yj) and the average of T (un)(x, ∆t
2 ) on Ij− 12
=
[xj−1, xj ] by the formula (3.10). Construct the Hermite type reconstructionpolynomials pj− 1
2(x) on the staggered interval Ij− 1
2= [xj−1, xj ] using the
formulas in Section 2.4. Let uj denote un(yj). Evaluate the extrema of pj− 1
2(x) in each staggered in-
terval Ij− 12
and the extrema of rj(x) in each Ij to calculate V ar(pj− 12) =∫ xj
xj−1|p′
j− 12
(x)| dx + |pj− 12(xj−1) − uj−1| + |pj− 1
2(xj) − uj | and the stan-
dard variation of un(x) on [yj−1, yj ], TV (un)[yj−1,yj ] =∫ yj
yj−1|(un)′(x)| dx.
If V ar(pj− 12) > TV (un)[yj−1,yj ], we perform the TVD limiting process de-
scribed in Section 2 on pj− 12(x) :
• If un(x) is monotone on [yj−1, yj ], then perform the limiting procedure
pj− 12(x) = pj− 1
2(x) − αj− 1
2(x − xj− 1
2),
where αj− 12
=
{minx∈I
j− 12
p′j− 1
2
(x), if uj−1 ≤ uj
maxx∈Ij− 1
2
p′j− 1
2
(x), if uj−1 ≥ uj, and
rj− 12(x) = θj− 1
2
(pj− 1
2(x) − un+1
j− 12
)+ un+1
j− 12
,
with θj− 12
= min
{∣∣∣∣∣uj−1−un+1
j− 12
epj− 1
2
(xj−1)−un+1
j− 12
∣∣∣∣∣ ,∣∣∣∣∣
uj−un+1
j− 12
epj− 1
2
(xj)−un+1
j− 12
∣∣∣∣∣ , 1}
.
• If un(x) is not monotone on [yj−1, yj ], then choose either the maximumor the minimum of un(x), denoted as yext
j ∈ [yj−1, yj ]. Set xextj =
yextj + f ′(un(yext
j ))∆t/2 and uextj = un(yext
j ). Divide Ij− 12
into two
parts I lj− 1
2
= [xj−1, xextj ] and Ir
j− 12
= [xextj , xj ]. Calculate the averages
12
of T (un)(x, ∆t2 ) over I l
j− 12
and Irj− 1
2
by
ulj− 1
2
= 1∆x
[∫ yextj
yj−1un(x)dx − ∆t
2 f(un(yextj )) + (xext
j − yextj )un(yext
j )
+∆t2 f(un(yj−1)) − (xj−1 − yj−1)u
n(yj−1)]
urj− 1
2
= 1∆x
[∫ yj
yextj
un(x)dx − ∆t2 f(un(yj)) + (xj − yj)u
n(yj)
+∆t2 f(un(yext
j )) − (xextj − yext
j )un(yextj )
].
Set plj− 1
2
(x) = pj− 12(x) and pr
j− 12
(x) = pj− 12(x). Let pl
j− 12
denote the
average of plj− 1
2
(x) on I lj− 1
2
. Do the following limiting procedure to
plj− 1
2
(x):
(a) If(uj−1 − ul
j− 12
)(ul
j− 12
− uextj
)< 0, set rl
j− 12
(x) = ulj− 1
2
;
(b) Else, if plj− 1
2
(x) is monotone and(pl
j− 12
(xj−1) − plj− 1
2
(xextj )
)(uj−1−
uextj ) < 0, then set rl
j− 12
(x) = ulj− 1
2
;
(c) Else, if plj− 1
2
(x) is monotone and(pl
j− 12
(xj−1) − plj− 1
2
(xextj )
)(uj−1−
uextj ) ≥ 0, then set pl
j− 12
(x) = plj− 1
2
(x)−plj− 1
2
+ulj− 1
2
, and rlj− 1
2
(x) =
θj− 12(pl
j− 12
(x) − ulj− 1
2
) + ulj− 1
2
, where
θj− 12
= min
{∣∣∣∣uj−1−ul
j− 12
epj− 1
2
(xj−1)−ul
j− 12
∣∣∣∣ ,
∣∣∣∣uext
j −ul
j− 12
epj− 1
2
(xextj
)−ul
j− 12
∣∣∣∣ , 1}
;
(d) Else, if plj− 1
2
(x) is not monotone, set
plj− 1
2
(x) = plj− 1
2
(x) − plj− 1
2
+ ulj− 1
2
− αj− 12
(x −
xj−1 + xextj
2
),
with αj− 12
=
minx∈Il
j− 12
(pl
j− 12
)′
(x), if uj−1 ≤ uextj
maxx∈Il
j− 12
(pl
j− 12
)′
(x), if uj−1 ≥ uextj
, and
rlj(x) = θj− 1
2(pl
j− 12
(x) − ulj− 1
2
) + ulj− 1
2
,
where θj− 12
= min
{∣∣∣∣uj−1−ul
j− 12
epl
j− 12
(xj−1)−ul
j− 12
∣∣∣∣ ,
∣∣∣∣uext
j −ul
j− 12
epl
j− 12
(xext)−ul
j− 12
∣∣∣∣ , 1}
.
Perform a similar modification to prj− 1
2
(x), then we obtain rj− 12(x) =
rlj− 1
2
(x)χ(I lj− 1
2
) + rrj− 1
2
(x)χ(Irj− 1
2
).
This finishes the evolution to tn + ∆t2 on the staggered mesh Ij− 1
2.
5. Evolve to tn + ∆t in an analogous way back to the mesh Ij .
4. Properties of the scheme.
4.1. Conservative form. Theorem 4.1. The scheme given in the previoussection can be written in a conservative form.
13
x
t
� � �
� �
� �
xj−1 xj
xj− 3
2
xj− 1
2
xj+ 1
2
tn
tn+1
2
yj−1 yj
Fig. 4.1. The trapezoid region in the x-t plane.
Proof. The cell averages can be rewritten as
un+ 1
2
j− 12
= 1∆x
[∫ yj
yj−1un(x)dx − ∆t
2 f(uj) + (xj − yj)uj + ∆t2 f(uj−1) − (xj−1 − yj−1)uj−1
]
= unj− 1
2
− 12
∆t∆x(fj − fj−1),
where the numerical flux is fj = 2∆t
∫ xj
yjun(x)dx +
(f(uj) −
2(xj−yj)∆t uj
)and un
j− 12
denotes the average of un(x) on Ij− 12. See Figure 4.1.
First, fj only depends on un(x) over the interval Ij . Second, fj is consistent in
the sense that fj = f(u) if un(x) is a constant u. Thus, our scheme is conservative.
4.2. Total-variation diminishing. Theorem 4.2. The scheme given in theprevious section is TVD: TV (un+ 1
2 (x)) ≤ TV (un(x)).Proof. As long as the cell averages satisfy the maximum principle (3.9), Lemma
2.2 in the TVD reconstruction section will hold, which ensures the TVD propertyof the reconstruction after limiting. The preconditioning process in Section 3.2 cleardoes not increase the variation. Therefore, the scheme satisfies
TV (un+ 12 (x)) ≤ V ar(un+ 1
2 (x)) ≤ TV (un(x)).
4.3. Accuracy. Except for the Case 3 in Lemma 3.2, the time evolution of ourscheme is exactly the same as [12]. Therefore, following the same lines as in [12],if Case 3 never happens, then we can show that our scheme is fifth order accuratefor smooth solutions away from the extrema by calculating the local truncation errordefined in [12], and it will lose at most one order of accuracy near extrema. Sincethere is only finitely many such extrema, the L1 order of accuracy is optimal. Here,we need to show that the local truncation error will lose at most one order of accuracyif Case 3 happens. It suffices to check the accuracy of the backward characteristicline that we choose in Case 3.
Lemma 4.3. Assume f ′′(x) and u′0(x) are both bounded, then all the three can-
didates of the backward characteristic lines in Case 3 are fifth order accurate if themesh is fine enough.
14
Proof. Assume the three possible locations of the foot of backward characteristicsare s0, s1 and s2 as shown in Figure 3.1(b) (in which s0 is denoted as x0
j ). It suffices
to show that s1 − s2 = O(∆x5) if ∆x and ∆t are sufficiently small.Define G(x) = f ′(u0(x))∆t + 2(x− xj) and H(x) = f ′(u0(x))∆t + 2(x− xj). We
have
G(x) − H(x) = f ′(u0(x))∆t − f ′(u0(x))∆t = f ′′(ζ)(u0(x) − u0(x))∆t = O(∆x5),
therefore |H(s−0 ) − H(s+0 )| ≤ |H(s−0 ) − G(s0)| + |G(s0) − H(s+
0 )| = O(∆x5), henceH(s−0 ) H(s+
0 ) < 0 implies |H(s−0 )| = O(∆x5). Notice that
G′(x) = f ′′(u0(x))u′0(x)∆t + 2,
hence we have G′(x) ≥ 32 for all x ∈ [xj− 1
2, xj+ 1
2] if ∆t is small enough, since f ′′(x)
and u′0(x) are bounded. Now G′(x) − H ′(x) = O(∆x4) implies that if ∆x is small,
H ′(x) ≥ 1 for all x ∈ [xj− 12, xj+ 1
2]. Therefore
H(s−0 ) − H(s1)
s0 − s1= H ′(ξ) ≥ 1, for some ξ ∈ [s1, s0],
which implies s0 − s1 ≤ H(s−0 ) − H(s1) = H(s−0 ) = O(∆x5). Similarly s0 − s2 =O(∆x5). Therefore, the accuracy is not destroyed even if the entropic characteristicline is not necessarily chosen in Case 3.
We now have the following theorem on the accuracy of our scheme.Theorem 4.4. Assuming the initial data u0(x) and the flux f(u) are both smooth
functions. Take ∆t sufficiently small so that (3.1) is satisfied. The scheme is fifthorder accurate away from the extrema of u0(x) and when Case 3 does not happen.Accuracy can lose at most one order near the extrema or when Case 3 happens, hencein L1 the error is fifth order accurate.
5. Numerical test. In this section we provide numerical examples to test ourschemes.
5.1. Standard test cases. Example 1. We solve the model equation ut+ux =0,−1 ≤ x ≤ 1, u(x, 0) = u0(x), with periodic boundary conditions.
Three initial data u0(x) are used. The first one is u0(x) = sin(πx), and the secondone is u0(x) = sin4(πx). We list the L1 and L∞ errors for the cell averages at timet = 5 in Table 5.1. Here and below, the mesh size ∆x = 2/N . We can clearly see thatthe designed fifth order accuracy is achieved in both cases, at least for the L1 error.
Table 5.1
t = 5, ∆t/∆x = 0.95.
N u0(x) = sin(πx) u0(x) = sin4(πx)L1 error order L∞ error order L1 error order L∞ error order
20 1.08E-6 – 2.18E-6 – 6.25E-4 – 1.22E-3 –40 3.34E-8 5.01 6.93E-8 4.97 2.72E-5 4.52 7.58E-5 4.0180 1.02E-9 5.03 2.17E-9 5.00 8.46E-7 5.00 3.15E-6 4.59160 3.14E-11 5.02 6.69E-11 5.02 2.65E-8 4.99 1.20E-7 5.13320 8.57E-10 4.95 4.08E-9 4.47
The third initial function is u0(x) =
{1, −1 ≤ x ≤ 0−1, 0 ≤ x ≤ 1
. The results at t = 100
are shown in Figure 5.2(a). We can see that numerical solution maintains a strict
15
maximum principle and has relatively good resolution for the discontinuity for a coarsemesh after a very long time simulation (50 time periods).
Example 2. We solve the Burgers equation with periodic boundary conditions ut +(u2/2
)x
= 0,−1 ≤ x ≤ 1, u(x, 0) = u0(x). For the initial data u0(x) = 0.25 +
0.5 sin(πx), the exact solution is smooth up to t = 2π , then it develops a moving shock
which interacts with a rarefaction wave. We list the errors in Table 5.2 at t = 0.15.We can clearly see the designed fifth order accuracy is achieved in the L1 norm. InFigure 5.1 we can see that the shock is captured very well at t = 2
π and t = 2.0.The errors 0.05 away from the shock (i.e., |x− shock location| ≥ 0.05) are listed inTable 5.2 at t = 2.0. We can see that the designed order of accuracy is achieved orsurpassed.
Table 5.2
u0(x) = 0.25 + 0.5 sin(πx), ∆t/∆x = 1.2.
N t = 0.15 t = 2L1 error order L∞ error order L1 error order L∞ error order
20 1.32E-6 – 6.73E-6 – 2.19E-3 – 4.09E-2 –40 3.26E-8 5.34 1.47E-7 5.51 3.30E-5 6.05 7.34E-4 5.8080 6.41E-10 5.66 4.25E-9 5.12 2.68E-6 6.95 1.79E-5 5.36160 2.30E-11 4.80 2.38E-10 4.16 8.23E-10 8.34 9.24E-8 9.24320 7.63E-13 4.91 1.59E-11 3.90 2.70E-13 11.57 9.30E-12 13.27640 2.46E-14 4.95 1.02E-12 3.95
x
u(x)
-0.5 0 0.5
-0.2
0
0.2
0.4
0.6
(a) t = 0.6366
x
u(x
)
-0.5 0 0.5-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(b) t = 2
Fig. 5.1. N = 80, ∆t/∆x = 1.2. Solid line: exact solution; Symbols: numerical solution.
Example 3. Note that we have only proved TVD for our schemes, not entropy con-ditions. We use nonconvex fluxes to test the convergence to the physically correct en-tropy solutions. The “exact” solutions are obtained from the first order Lax-Friedrichsscheme on a very fine mesh.
The first flux is the Buckley-Leverett flux f(u) = 4u2
4u2+(1−u)2 , with the initial data
u = 1 in [− 12 , 0] and u = 0 elsewhere. The computational result is displayed in Figure
5.2(b), which is quite satisfactory.
16
The second flux is f(u) =
1, if u < 1.6cos(5π(u − 1.8)) + 2.0, if 1.6 ≤ u < 2.0− cos(5π(u − 2.2)), if 2.0 ≤ u < 2.41, if u ≥ 2.4
with
two initial conditions, which is an example used in [11]. The first initial condition is
u0(x) =
{1, for x < 03, for x ≥ 0
and it is shown in [11] that the numerical solutions of
many high order schemes would stay stationary which is entropy-violating. Our resultis shown in Figure 5.2(c) (solid line is exact solution and symbols denote numericalsolution (cell averages)) which approximates the exact entropy solution very well. The
other initial data that we test is u0(x) =
{3, for − 1 ≤ x < 01, for 0 ≤ x ≤ 1
with a periodic
boundary condition. It is shown in [11] that convergence towards the entropy solutionfor this test case is slow for first order monotone schemes and may fail for many highorder schemes. Our results are shown in Figure 5.2(d). There is clearly convergencewith refined meshes, and the rate of convergence is faster than that of the first orderschemes shown in [11].
5.2. Test cases from traffic flow models. In the subsection, we test ourfifth order TVD scheme on two traffic flow problems. To describe the dynamiccharacteristics of traffic on a homogeneous and unidirectional highway, the Lighthill-Whitham-Richards (LWR) model is widely used. The equation for the LWR modelis ρt + q(ρ)x = 0 with suitable initial and boundary conditions. Here ρ ∈ (0, ρmax) isthe density, ρmax is the maximum (jam) density, and q(ρ) = u(ρ)ρ is the traffic flowon a homogeneous highway.Example 4. The first traffic flow test example is taken from [7]. The flow-densityfunction is given by a concave function
q(ρ) =
−0.4ρ2 + 100ρ, ρ ∈ [0, 50]−0.1ρ2 + 15ρ + 3500, ρ ∈ [50, 100]−0.024ρ2 − 5.2ρ + 4760, ρ ∈ [100, 350]
.
The length of the freeway is 20 km. The entrance density is 50 veh/km. The piecewiselinear initial density profile shown in Fig 5.3(a) is formed. The entrance is blockedfor 10 min, after which traffic is released again from the entrance at the capacitydensity 75 veh/km. After 20 min, the entrance flow returns to 50 veh/h. At the exitboundary, a traffic signal is installed, with a repeated pattern of 2 min green light(zero density) followed by 1 min red light (jam density). The numerical solutions areshown in Figures 5.3(b), 5.3(c) and 5.3(d). We can observe that our TVD schemeproduces very good approximations to the exact solution for this test case.Example 5. We consider a similar problem but with a much more complicatedflow-density function in [3]. The flow function q(ρ) = ρVe(ρ) is given by
Ve(ρ) =eV 2
2V0
(−1 +
√1 +
4V 20
eV 2
)
with V (ρ) = 1Tr
(1ρ − 1
ρmax
)√α(ρmax)
α(ρ) and α(ρ) = α0 + ∆α(tanh (ρ−ρc
∆ρ ) + 1)
. Here
V0, Tr, ρmax, α0, ∆α, ρc and ∆ρ are all constant parameters to be determined byfitting them to the empirical data. The physical meaning of these parameters canbe found in [3]. We simply choose some typical values mentioned in [3]: V0 = 110km/h, Tr = 1.8 seconds, ρmax = 160 vehicles/km, α0 = 0.008, ∆α = 0.02, ρc =
17
x
u(x
)
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
(a) N = 40, t = 100.
x
u(x
)
-1 -0.5 0 0.5 1
0
0.2
0.4
0.6
0.8
1
(b) N = 160, t = 0.4, ∆t/∆x = 0.3.
x
u(x
)
-10 -5 0 5 10
1
1.5
2
2.5
3
(c) N = 400, t = 2, ∆t/∆x = 0.04.
x
u(x
)
-0.5 0 0.5 1
1.99
1.995
2
2.005
2.01
(d) t = 2, ∆t/∆x = 0.04. The curves, fromright to left, are corresponding to N = 800,N = 1600, N = 3200 and the “exact” solution,respectively.
Fig. 5.2. Example 1 and Example 3. In Fig 5.2(a), 5.2(b) and 5.2(c), solid line is exactsolution and squares denote numerical solution (cell averages).
0.27ρmax and ∆ρ = 0.1ρmax. With all these parameters, the flow-density functionq(ρ) is well-defined, the graphs of this function and its second derivative are plottedin Figure 5.4(a) and 5.4(b). It is clearly neither a globally concave nor a globallyconvex function. The entrance density is constant 30 veh/km. The initial conditionis ρ0(x) = 135
2 sin ( π10x) + 145
2 . At the exit boundary, a traffic signal is installed, witha repeated pattern of 1 min green light (ρ = 10 veh/km) followed by 2 min redlight (ρ = 140 veh/km). The numerical solutions are shown in Figure 5.4(c), witha magnified graph for the boxed region shown in Figure 5.4(d), where the solid lineis the reference solution obtained by the first order Lax-Friedrichs scheme on a veryfine grid (N = 4000000) at t = 18 min. We again observe very good resolution of ourscheme for this nonconvex traffic flow model.
Remark 5.1. The advantage of our scheme is that it is high order accurate andsatisfies strictly a maximum principle, therefore it has better resolution than the usual
18
Distance (km)
Den
sity
(veh
/km
)
5 10 15
50
100
150
200
250
300
350
(a) The initial density.
Distance
Den
sity
0 5 10 15 200
50
100
150
200
250
300
350
(b) t = 30 min.
Distance
Den
sity
0 5 10 15 200
50
100
150
200
250
300
350
(c) t = 60 min.
Distance
Den
sity
0 5 10 15 200
50
100
150
200
250
300
350
(d) t = 90 min.
Fig. 5.3. Traffic flow Example 4: N = 800. Solid line: exact solution; Circles: numericalsolution (cell averages).
TVD schemes when the solution contains many waves, and it does not generate anyunphysical solution such as negative density. From the two examples above, we cansee that all the waves (shocks and rarefactions) are well captured.
Remark 5.2. The boundary conditions in these two examples are all piecewiseconstants in time. Hence we simply use constant values on ghost cells as the numericalboundary condition for our scheme.
5.3. A simplified scheme. In this subsection we discussed a simplified versionof our finite volume scheme. We use the same method for time evolution, and asimilar but simpler limiter which only enforces strict maximum principle but notTVD, without breaking a polynomial into two pieces on any interval. The algorithmsatisfies all the theoretical properties of the previous scheme except for a rigorousproof of TVD. This maximum-principle-satisfying finite volume scheme is describedbelow.
1. Start with the preconditioned version of piecewise polynomial of degree fourun(x) =
∑j rj(x)χj(x) at time level n.
19
Density (veh/km)
Flo
w(v
eh/h
)
0 50 100 1500
500
1000
1500
2000
(a) the graph of q(ρ).
0 50 100 150
-20
-15
-10
-5
0
5
10
15
(b) the graph of q′′(ρ).
Distance
Den
sity
0 5 10 15 200
20
40
60
80
100
120
140
160
(c) t = 18 min. The region inside the rectangleon the right is magnified in Figure 5.4(d).
Distance
Den
sity
17 18 19 200
20
40
60
80
100
120
140
160
(d) Magnified graph of the region inside therectangle in Figure 5.4(c).
Fig. 5.4. Traffic flow Example 5: N = 1600. Solid line: exact solution; Circles: numericalsolution (cell averages).
2. For each xj , find the unique root yj of the characteristic equation (3.4) in Ij .The uniqueness of the root is ensured by Lemma 3.1.
3. Evaluate T (un)(xj ,∆t2 ) = un(yj) and the average u
n+ 12
j− 12
of T (un)(x, ∆t2 ) on
Ij− 12
by the formula (3.3). Construct the Hermite type interpolation polyno-
mials pj− 12(x) on the staggered interval Ij− 1
2= [xj−1, xj ].
4. For each interval Ij− 12, evaluate the maximum Mj and the minimum mj of
pj− 12(x) on Ij− 1
2, and the maximum Mj and the minimum mj of un(x) on
[yj−1, yj ]. Apply the following scaling rj− 12(x) = θ(pj− 1
2(x) − u
n+ 12
j− 12
) + un+ 1
2
j− 12
where θ is determined by θ = min
fMj−un+1
2
j− 12
Mj−un+1
2
j− 12
,emj−u
n+12
j− 12
mj−un+1
2
j− 12
, 1
.
5. Apply the preconditioning process. This finishes the evolution to tn + ∆t2 on
the staggered mesh Ij− 12.
20
6. Evolve to tn + ∆t in an analogous way back to the mesh Ij .
Remark 5.3. We can easily prove that this scheme is conservative and fifth orderaccurate. Obviously it satisfies the maximum principle. Therefore we will refer to itas the maximum-principle-satisfying scheme.
Remark 5.4. The maximum-principle-satisfying scheme is much easier to codethan the TVD scheme. Although we cannot rigorously prove TVD or TVB of the nu-merical solution, we have not observed any significant difference between this schemeand the TVD scheme tested before, for all the test cases reported in this paper. Wewill not show these results here to save space.
6. Concluding remarks. In this paper we have extended the work in [12] andhave constructed a class of genuinely high order accurate finite volume TVD schemesfor solving one dimensional scalar conservation laws. These schemes do not degen-erate to lower order accuracy for solutions with smooth extrema, yet they satisfy astrict maximum principle and they can be proved to be TVD, when the total varia-tion is measured by the bounded variation semi-norm of the reconstructed piecewisepolynomials. The key ingredient of the algorithm is the TVD reconstruction, whichcan be efficiently implemented for order of accuracy up to six.
We have tested the fifth order scheme on a variety of examples including thosefrom traffic flow models and those with non-convex fluxes. The solutions are highorder accurate and provide good resolution to shocks and rarefaction waves.
A simplified scheme is also described, which enforces a strict maximum principlebut is not rigorously TVD, however it is much simpler to implement than the TVDscheme. Numerical experiments indicate that this simplified scheme perform as nicelyin all the test cases as the TVD scheme.
The advantage of the schemes constructed in this paper, compared with tradi-tional TVD schemes, is that they do not degenerate to first order at smooth extrema,hence they give very good resolutions to solutions with complicated smooth waves.On the other hand, the advantage of the schemes constructed in this paper, comparedwith ENO and WENO schemes, is that they satisfy strictly the maximum principle,hence they will not generate non-physical solutions such as negative density for thetraffic flows. This property is important in many applications.
In order to generalize of this scheme to two dimensions, we should abandon therequirement of TVD and insist only on the strict maximum principle, that is, along theapproach of the simplified scheme in Section 5.3. Initial work along this direction hasbeen performed in [16], after the submission of the original version of this paper. It isalso possible to formally generalize the scheme to hyperbolic systems, see [13] for onepossible approach. However, it would be probably more appropriate, for the systemcase, to enforce certain positivity preserving properties, such as positivity preservingfor density and pressure for Euler equations of compressible gas dynamics, see [9] forone possible approach. These generalizations constitute ongoing research.
REFERENCES
[1] A. Harten, High resolution schemes for hyperbolic conservation laws, Journal of ComputationalPhysics, 49 (1983), pp.357-393.
[2] A. Harten, B. Engquist, S. Osher and S. Chakravarthy, Uniformly high order essentially non-oscillatory schemes, III, Journal of Computational Physics, 71 (1987), 231-303.
[3] D. Helbing, A. Hennecke, V. Shvetsov and M. Treiber, MASTER: macroscopic traffic simula-tion based on a gas-kinetic, non-local traffic model, Transportation Research Part B, 35(2001), pp.183-211.
21
[4] G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes, Journal ofComputational Physics, 126 (1996), pp.202-228.
[5] X.-D. Liu and S. Osher, Non-oscillatory high order accurate self similar maximum princi-ple satisfying shock capturing schemes, SIAM Journal on Numerical Analysis, 33 (1996),pp.760-779.
[6] X.-D. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes, Journal ofComputational Physics, 115 (1994), 200-212.
[7] Y. Lu, S.C. Wong, M. Zhang, C.-W. Shu and W. Chen, Explicit construction of entropy so-lutions for the Lighthill-Whitham-Richards traffic flow model with a piecewise quadraticflow-density relationship, Transportation Research Part B, 42 (2008), pp.355-372.
[8] S. Osher and S. Chakravarthy, High resolution schemes and the entropy condition, SIAMJournal on Numerical Analysis, 21 (1984), pp.955-984.
[9] B. Perthame and C.-W. Shu, On positivity preserving finite volume schemes for Euler equa-tions, Numerische Mathematik, 73 (1996), pp.119-130.
[10] J.-M. Qiu and C.-W. Shu, Convergence of Godunov-type schemes for scalar conservation lawsunder large time steps, SIAM Journal on Numerical Analysis, 46 (2008), pp.2211-2237.
[11] J.-M. Qiu and C.-W. Shu, Convergence of high order finite volume weighted essentially non-oscillatory scheme and discontinuous Galerkin method for nonconvex conservation laws,SIAM Journal on Scientific Computing, 31 (2008), pp.584-607.
[12] R. Sanders, A third-order accurate variation nonexpansive difference scheme for single non-linear conservation law, Mathematics of Computation, 51 (1988), pp.535-558.
[13] R. Sanders, High resolution staggered mesh approach for nonlinear hyperbolic systems of con-servation laws, Journal of Computational Physics, 101 (1992), pp.314-329.
[14] C.-W. Shu, TVB uniformly high-order schemes for conservation laws, Mathematics of Com-putation, 49 (1987), pp.105-121.
[15] C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of Computational Physics, 77 (1988), pp.439-471.
[16] X. Zhang and C.-W. Shu, On maximum-principle-satisfying high order schemes for scalarconservation laws, Journal of Computational Physics, 229 (2010), pp.3091-3120.
Appendix A. Appendix.
In this appendix we provide a proof of Lemma 2.3.Proof: We only prove the k = 4 case. The proof for lower k is very similar (and
simpler) and is thus omitted.We can assume the interval is I = [− 1
2 , 12 ] by considering the rescaled variable
x′ = x−(a+b)/2b−a , which leaves the ratios in (2.7) unchanged. We will still denote x′ by
x and, without loss of generality, we assume p(x) = a0 + a1x + a2x2 + a3x
3 + a4x4
is increasing on I = [− 12 , 1
2 ]. Notice that since p(x) is monotone on I = [− 12 , 1
2 ], weonly need to show that
(A.1)
∣∣∣∣∣p
(12
)− p
p(− 1
2
)− p
∣∣∣∣∣ ≤ 7,
∣∣∣∣∣p
(− 1
2
)− p
p(
12
)− p
∣∣∣∣∣ ≤ 7.
We will only prove the first inequality in (A.1), the derivation for the secondinequality being the same. We have
p(− 1
2
)= a0 −
12a1 + 1
4a2 −18a3 + 1
16a4, p(
12
)= a0 + 1
2a1 + 14a2 + 1
8a3 + 116a4
p = a0 + 112a2 + 1
80a4
Thus,
∣∣∣∣∣p
(12
)− p
p(− 1
2
)− p
∣∣∣∣∣ =p( 1
2 ) − p
−p(− 12 ) + p
=12a1 + 1
6a2 + 18a3 + 1
20a4
12a1 −
16a2 + 1
8a3 −120a4
= 1 +23a2 + 1
5a4
a1 −13a2 + 1
4a3 −110a4
= 1 +2
3
a2 + 310a4
a1 −13a2 + 1
4a3 −110a4
.
22
We now discuss this in several cases.
Case 1. If a2 + 310a4 = 0, then
∣∣∣∣p( 1
2 )−p
p(− 12 )−p
∣∣∣∣ = 1 < 7.
Case 2. If a2 + 310a4 < 0, then
∣∣∣∣p( 1
2 )−p
p(− 12 )−p
∣∣∣∣ = 1 + 23
1a1+ 1
4a3
a2+ 310
a4
− 13
. Since p(x) is
increasing,
1
2a1 +
1
6a2 +
1
8a3 +
1
20a4 = p
(1
2
)− p ≥ 0,
=⇒1
2(a1 +
1
4a3) +
1
6(a2 +
3
10a4) ≥ 0,
=⇒a1 + 1
4a3
a2 + 310a4
≤ −1
3,
=⇒
∣∣∣∣∣p
(12
)− p
p(− 1
2
)− p
∣∣∣∣∣ < 1.
Case 3. If a2 + 310a4 > 0, then
∣∣∣∣∣p
(12
)− p
p(− 1
2
)− p
∣∣∣∣∣ = 1 +2
3
1a1+ 1
4a3
a2+310
a4− 1
3
.
For any u, v ∈ [− 12 , 1
2 ], we have
p′(u) = a1 + 2ua2 + 3u2a3 + 4u3a4 ≥ 0,
p′(v) = a1 + 2va2 + 3v2a3 + 4v3a4 ≥ 0,
=⇒ p′(u) + p′(v) = 2a1 + 2(u + v)a2 + 3(u2 + v2)a3 + 4(u3 + v3)a4 ≥ 0,
=⇒ 2(a1 +3
2(u2 + v2)a3) + 2(u + v)(a2 + 2(u2 + v2 − uv)a4) ≥ 0,(A.2)
We would like to find u, v ∈ [− 12 , 1
2 ] satisfying
{32 (u2 + v2) = 1
42(u2 + v2 − uv) = 3
10
.
Solving this linear system, we obtain u = − 12
(√15 +
√215
)and v = − 1
2
(√15 −
√215
),
which are apparently within [− 12 , 1
2 ]. Plugging these values into (A.2), we obtain
2
(a1 +
1
4a3
)− 2
√1
5
(a2 +
3
10a4
)≥ 0,
=⇒a1 + 1
4a3
a2 + 310a4
≥
√1
5,
=⇒
∣∣∣∣∣p
(12
)− p
p(− 1
2
)− p
∣∣∣∣∣ ≤ 1 +2
3
1√15 − 1
3
≈ 6.85 < 7.
Remark A.1. The result of this lemma also holds for polynomials of higherdegree. We refer to Lemma 2.4. in [16] for the existence proof of the constant C. Theproof in [16] does not however provide an explicit value of C.
23